Kazimierz Zarankiewicz (2 May 1902 – 5 September 1959) was a Polish

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

and Professor at the

Warsaw University of Technology The Warsaw University of Technology ( pl, Politechnika Warszawska, lit=Varsovian Polytechnic) is one of the leading institutes of technology in Poland and one of the largest in Central Europe. It employs 2,453 teaching faculty, with 357 professor ...

who was interested primarily in

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

and

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

Biography

Zarankiewicz was born in

Częstochowa Częstochowa ( , ; german: Tschenstochau, Czenstochau; la, Czanstochova) is a city in southern Poland on the Warta River with 214,342 inhabitants, making it the thirteenth-largest city in Poland. It is situated in the Silesian Voivodeship (admin ...

. He studied at the

University of Warsaw The University of Warsaw ( pl, Uniwersytet Warszawski, la, Universitas Varsoviensis) is a public university in Warsaw, Poland. Established in 1816, it is the largest institution of higher learning in the country offering 37 different fields of ...

, together with

Zygmunt Janiszewski Zygmunt Janiszewski (12 July 1888 – 3 January 1920) was a Polish mathematician. Early life and education He was born to mother Julia Szulc-Chojnicka and father, Czeslaw Janiszewski who was a graduate of the University of Warsaw and served as t ...

Stefan Mazurkiewicz Stefan Mazurkiewicz (25 September 1888 – 19 June 1945) was a Polish mathematician who worked in mathematical analysis, topology, and probability. He was a student of Wacław Sierpiński and a member of the Polish Academy of Learning (''PAU''). ...

Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and to ...

Kazimierz Kuratowski Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Biography and studies Kazimierz Kuratowski was born in Warsaw, (th ...

, and Stanisław Saks. During

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...

, Zarankiewicz took part in illegal teaching, forbidden by the

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ger ...

authorities, and eventually was sent to a

concentration camp Internment is the imprisonment of people, commonly in large groups, without charges or intent to file charges. The term is especially used for the confinement "of enemy citizens in wartime or of terrorism suspects". Thus, while it can simply ...

. He survived and became a teacher at

. He visited universities in

Tomsk Tomsk ( rus, Томск, p=tomsk, sty, Түң-тора) is a city and the administrative center of Tomsk Oblast in Russia, located on the Tom River. Population: Founded in 1604, Tomsk is one of the oldest cities in Siberia. The city is a not ...

Harvard Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...

London London is the capital and largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River Thames in south-east England at the head of a estuary down to the North Sea, and has been a majo ...

, and

Vienna en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST ...

. He served as president of the

Warsaw Warsaw ( pl, Warszawa, ), officially the Capital City of Warsaw,, abbreviation: ''m.st. Warszawa'' is the capital and largest city of Poland. The metropolis stands on the River Vistula in east-central Poland, and its population is officia ...

section of the

Polish Mathematical Society The Polish Mathematical Society ( pl, Polskie Towarzystwo Matematyczne) is the main professional society of Polish mathematicians and represents Polish mathematics within the European Mathematical Society (EMS) and the International Mathematical Un ...

and the

International Astronautical Federation The International Astronautical Federation (IAF) is an international space advocacy organization based in Paris, and founded in 1951 as a non-governmental organization to establish a dialogue between scientists around the world and to lay t ...

. He died in

England England is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe b ...

Research contributions

Zarankiewicz wrote works on

cut-point In topology, a cut-point is a point of a connected space such that its removal causes the resulting space to be disconnected. If removal of a point doesn't result in disconnected spaces, this point is called a non-cut point. For example, every poi ...

s in

connected space In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties tha ...

s, on

conformal mapping In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...

s, on

complex functions Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...

and

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, and

triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...

s. The

Zarankiewicz problem The Zarankiewicz problem, an unsolved problem in mathematics, asks for the largest possible number of edges in a bipartite graph that has a given number of vertices and has no complete bipartite subgraphs of a given size.. Reprint of 1978 Academi ...

is named after Zarankiewicz. This problem asks, for a given size of

(0,1)-matrix A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. Matrix representati ...

, how many matrix entries must be set equal to 1 in order to guarantee that the matrix contains at least one ''a'' × ''b''

submatrix In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin ...

is made up only of 1's. An equivalent formulation in

extremal graph theory Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local ...

asks for the maximum number of edges in a

bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are ...

with no complete bipartite subgraph ''K''_''a'',''b''. The

Zarankiewicz crossing number conjecture Kazimierz Zarankiewicz (2 May 1902 – 5 September 1959) was a Polish mathematician and Professor at the Warsaw University of Technology who was interested primarily in topology and graph theory. Biography Zarankiewicz was born in Częstochowa. ...

in the mathematical field of

is also named after Zarankiewicz. The

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

states that the crossing number of a

complete bipartite graph In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory ...

K_

equals :

\text(K_) = \left\lfloor\frac\right\rfloor\left\lfloor \frac\right\rfloor\left\lfloor \frac\right\rfloor\left\lfloor \frac\right\rfloor.\

Zarankiewicz proved that this formula is an upper bound for the actual crossing number. The problem of determining the number

cr(K_)

was suggested by

Paul Turán Paul may refer to: *Paul (given name), a given name (includes a list of people with that name) *Paul (surname), a list of people People Christianity * Paul the Apostle (AD c.5–c.64/65), also known as Saul of Tarsus or Saint Paul, early Chri ...

and became known as Turán's brick factory problem.

External links

* {{DEFAULTSORT:Zarankiewicz, Kazimierz 20th-century Polish mathematicians Topologists University of Warsaw alumni Warsaw University of Technology faculty Nazi concentration camp survivors People from Częstochowa 1902 births 1959 deaths

Biography

Research contributions

See also

External links